Scorpius in Arithmancy
by Drollittle
Summary: What is the perpendicular capacity principle, why doesn't it apply to the Sorting Hat, and when will things start looking more positive?


Scorpius went over yesterday's formulas in his head as he entered Arithmancy. _charm propulsion force=(wand density x wand length) + (intensity of intent) over (total degrees of angles in the wand core-180)._.. He looked deliberately away from Rose as he walked past her desk, gave Albus a half-smile and then sat down and took out his homework. They sat like that in most of the classes they had together: Rose in front, then Al, then Scorpius in the back. That way both Rose and Scorpius could talk to Al without talking to each other.

"Practice problems and essays!" Professor Weasley called, and parchment flew to her desk from the desks of all seven of Hogwarts' 7th year Arithmancy NEWT students. _The few, the persistent, the geeky, we seven_... Scorpius thought. As small as a class of seven seemed, it was still the largest 7th year Arithmancy class Hogwarts had had in a long time. People just didn't stick around for this stuff very often. Finding teachers for it was difficult too—Professor Vector had been wanting to retire for years and this year Hermione Weasley had finally agreed to leave her office at the ministry to fill in for a few years.

Professor Weasley stood. "On Tuesday we learned why a straight line, or 180 degree angle, is the most effective angle for projecting and propelling magic. We discussed wands and a few other examples. Mr. Poe, do you remember the prime factorization of 180?"

Ed Poe's mouth hung open for a second. "Yeah, it was 2, 2, 3, 3, uh... 5."

"And?" Professor Weasley prompted.

"...and infinite ones."

"Right. Mr. Yang, the formula for charm propulsion force..."

She didn't finish her sentence because Jack Yang had his wand out and there was a scratching sound on the chalkboard behind her with each symbol that Yang formed with a flying bit of chalk.

cpf=(m/v x l + int.) / (sum(deg)-180)

"Therefore," Yang summarized in his nasally voice, "straighter, more Ka-pow."

Next to Yang, Dahlia Dippet bit her lip. "...Divide by...so, the closer the wand core angle is to exactly 180 degrees, the more force it can have to propel charms?"

"That's what I said, Dah." Said Yang.

"Dah" Al repeated, laughing and drawing laughs from most of the class, then Al softened the hurt look on Dippet's face with, "it's ok, Dahlia, you said it more eloquently."

"Shh" hushed Professor Weasley. "Thank you, Yang, ten points to Ravenclaw for remembering the full formula correctly, though you really needn't pull your wand out for everything. Out of curiosity, what were a few of the specific examples you wrote your essays about for 180 degree magical objects?"

A few hands went up. Scorpius had written about the golden fiddle bow of Johnny Clements, but he doubted anyone knew about it, so he didn't raise his hand.

"Mr. Potter?" Weasley called.

"The Sword of Noble Gryffindor!"

"Good one!" Whispered Divya Narayanan, the other Gryffindor in the class.

"Obvious one..." Scorpius countered, but only so Al would hear.

Weasley nodded at Al. "The Sword of Gryffindor, certainly. And Miss Dippet?"

"Professor, I wrote about dittany stalks, but I'm worried that wasn't specific enough, because it wasn't—like—one magical object, but I listed some examples of specific times when people have used them, and—"

"That is probably fine." Weasley cut her off. Dahlia Dippet was one of those Ravenclaws that worked and worried over schoolwork incessantly.

"Enough review. Today's lesson:" Professor Weasley announced, "the capacity to hold, or to store enchantments and magical information. What geometric angle will support the most magical capacity? Let's do a simple experiment to get an idea."

She put on her desk four wood objects, each with a flat base and a thin stalk attached to one side that pointed upward, identical except that each one pointed up at a different angle.

Then she pointed to a box on the floor that held small cloth sacks.

"Those are filled with heavy sand. I need four of you...Poe, Malfoy, Narayanan, and Yang..." (She was choosing the students that participated the least) "...to each choose one of the angle models, and balance sandbags on top of them one by one until they fall or break."

Coming up from the back, Scorpius had last pick, and had to take the model that was sure to lose first. Its stalk pointed backward, at about a 120 degree angle from the base, so the first bag he set on top of it made it fall over.

"Mr. Malfoy has shown us that obtuse angles, this one 120 degrees, support the lowest capacity. Watch the others." Said Weasley.

Jack Yang levitated five sand bags from the box on to the end of his hairpin-angled model before it cracked at the corner and broke. "Snappy snappy," he said.

"The 10 degree angle failed. Go ahead and repair the model, please, Mr. Yang, before you put it back in the cupboard...and Mr. Malfoy, put yours away and return to your seat as well. Now we have Mr. Poe's 60 degree angle and Miss Narayanan's 90 degree angle. I suppose you can guess which one will be the strongest?"

There was a murmur of "90 degrees," and Scorpius remembered a rhyme from his childhood; _'A thousand years this plenty keep, Our corners square; not low nor steep.' _He leaned forward in his seat and whispered it to Al.

"Where did you hear that?"

"Don't remember."

As Poe added weight to his model, it did indeed break, and Narayanan's was left holding the most sandbags.

Weasley explained, "And so you see that the right angle proved strongest. We call this the perpendicular capacity principle, and we use it to predict the amounts of charms or enchantments that can be put on objects, without being weakened or warped over time. Tell me some objects in which you might see the perpendicular capacity principle."

Dippet raised her hand. "A box, Professor?"

"Yes, and there have been many enchanted boxes over the ages. Mr. Poe?"

"A mirror..." He said morosely.

"A square, or rectangular mirror. Yes. Miss Narayanan?"

"A doorway or gate."

"Good. Now, try thinking outside the—forgive the pun—outside the box a little."

Rose raised her hand and Scorpius watched the waves of her red hair as she spoke. "A goblet or grail would have more capacity than a regular cup, because the stem creates right angles with the tangent of the bottom of the cup as well as with the base of the goblet."

It sounded so sophisticated,_ 'the tangent of the bottom of the cup as well as with the base...' _Scorpius remembered having once told Rose that her terminology was pretty, and she had laughed. That was over now.

Suddenly Al's hand shot up.

"Yes Mr. Potter?"

"Sorry, Professor, but what about the Sorting Hat? I can't think of anything that has held more enchantment for longer, but it doesn't have a single right angle."

"Ah, no, it doesn't. The perpendicular capacity principle is a true principle, but it is not the _only_ true principle. You are getting a bit ahead of the class; the sorting hat is, that is to say, was, a cone. A cone includes a 180 degree line on top of a circle. We will discuss next week that circles have the power to recall, to remember. Lines, as we know, have the power to propel. And so, the Sorting Hat remembers and directs. Over time it has become wrinkled and frayed, giving it creativity of its own. A geometric study of the Hat would be complicated and very interesting, but let's get back to right angles. Can anyone think of an object that contains many right angles?"

"A net or screen?" Asked Yang.

"Hmm, I think a net is too fluid, the angles would not always be 90 degrees, but good thinking, Mr. Yang."

Scorpius thought...a piano. He smiled to himself but didn't raise his hand. No one else loved the piano like he did. What else? a quilt? a bookshelf? Then he thought of something Grandfather Malfoy had once mentioned. 'It was just the beginning of how the Dark Lord wronged our family.' Grandfather had said bitterly. Scorpius raised his hand.

"Mr. Malfoy?"

"A book, Professor."

Both Al and Rose turned and looked at Scorpius as they realized what he was talking about.

Weasley nodded. "A book has four right angles on each side of the cover, the spine, each page. Per each hundred pages, four hundred right angles. The text is generally set perpendicular to the side of the page also, and how many lines of text are on each page? You can see how it compounds. Add to that a book's ability to be written in, and I believe a book has the greatest capacity to store magic of any small object."

There was no 'good thinking, Mr. Malfoy. Ten points to Slytherin,' but Scorpius didn't care much.

"More ideas?" Weasley prompted. "Mr. Potter?"

"How about a building, like a castle?"

"Definitely. A castle, a pyramid...Wizards have been using the right angles of our structures to hold powerful enchantments for ages. Perhaps the most famous example of the intentional use of this principle is the Malfoy Manor." Weasley said, giving Scorpius a brief nod. He nodded back, but his nerves jumped to alert, as they did whenever anyone started talking about Malfoy history. "The eleventh century architecture uses a complex series of carefully composed right angles, enchanted in a design to retain prosperity, power, wealth..."

Rose tisked loudly.

"Not the most important things in life, we all know." Scorpius snapped.

"Five points from both Ravenclaw and Slytherin—compose yourselves." Said Weasley coolly, and continued, "History has shown the lasting strength of the Malfoy's enchantments, whether for better or worse. Those are sufficient examples for now; you'll need to think of others for your homework. Now let's get into the reasoning. Why would this perpendicular capacity principle apply to objects holding enchantments?"

The class was silent. Scorpius tried to think...why would an object with right angles hold magic better or longer than any other object?

Al raised his hand.

"I think it must be the balance, Aunt—Professor. Perpendicular lines have 90 degrees on all corners, and a grouping of 90 degree angles will automatically match up with each other. That balance, having the same value on both sides, would mean that all the strain goes directly down into the object itself rather than pushing to one side or the other. A large amount of power could be piled together. Over time, the charm or enchantment would just settle in, not wear out."

"Exactly, Mr. Potter. Ten points to Gryffindor." Weasley said proudly. Then Rose raised her hand as well.

"Just to add to what Al said, when you graph two perpendicular lines, the slopes of the lines will always be the negative reciprocals of each other." Rose leaned forward, explaining eagerly, "For example, the line y=2x forms right angles with the line y=-1/2x, and y=2/5x forms right angles with y=-5/2x. That shows the balance; top, bottom, positive, negative. If you multiply them together, they always equal negative one. In perfect balance like that, an enchantment on an object, even a top-heavy enchantment, would never be pulled too far to one side or the other. It would be stable."

Why did she have to be such a genius?

Weasley smiled, trying not to look too pleased with her own daughter. "Ten points for Ravenclaw too. Everyone take out graphing parchment; we are going to plot a few perpendicular lines so you can see the negative reciprocal slopes."

When they had finished that and Weasley had assigned homework, class was dismissed. Scorpius put his things away slowly while Rose left quickly.

He left the classroom, but something kept bothering Scorpius... _Never pulled too far to one side or the other. If you multiply them together, they always equal negative one. A thousand years this plenty keep, Our corners square; not low nor steep. Eleventh century architecture. _

Scorpius did not want to be another negative one. Maybe the thousand years was up. Maybe it was time to throw things out of balance.

**Notes:**

**This is part of a much longer second generation story I have in my head, but as a turning point for Scorpius I thought it would make a good one-shot. **

**The more I have thought about the possibilities in Arithmancy, the more I think the content here would be just 4th or 5th year material. **

happy=readers+reviews


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